Frequency transformation from a lowpass to a bandpass filter in s-domain.
Example 1
Design an analog bandpass filter, where the passband is defined with Wl = 0.2 rad/sec, Wu = 0.6 rad/sec
uses MtxExpr, Math387, MtxVec, MtxVecTee, MtxVecEdit,
LinearSystems, IirFilters, SignalUtils;
procedure TForm1.Button1Click(Sender: TObject);
var z,p, num,den, FreqFr,Response: Vector;
Order: integer;
k,Wc,BW: TSample;
begin
Order := 5; //design a fifth order filter.
EllipticAnalog(Order,0.2,40,z,p,k); //design analog protype
Wc := Sqrt(0.2*0.6);
BW := 0.6-0.2;
LowpassToBandpass(z,p,k,Wc,BW); //frequency transformation in s-domain
ZeroPoleToTransferFun(num,den,z,p,k);
//Define the frequency grid (logarithmic)
FreqFr.Length := 1000;
LogRamp(FreqFr,-1,1); //between 0.1 (=10^(-1)) and 10 (=10^1) rad/sec
FrequencyResponseS(num,den,FreqFr,Response); //Laplace
DrawIt(Response); //Y axis linear, X axis logarithmic
end;
#include "MtxVecCPP.h" //MtxVecCPP.cpp must be included in the project
#include "MtxVecEdit.hpp"
#include "MtxVecTee.hpp"
#include "SignalUtils.hpp"
#include "IirFilters.hpp"
#include "LinearSystems.hpp"
void __fastcall TForm1::BitBtn1Click(TObject *Sender)
{
Vector z,p, num,den, FreqFr,Response;
int Order;
double k,Wc,BW;
Order = 5; //design a fifth order filter.
EllipticAnalog(Order,0.2,40,z,p,k); //design analog protype
Wc = Sqrt(0.2*0.6);
BW = 0.6-0.2;
LowpassToBandpass(z,p,k,Wc,BW); //frequency transformation in s-domain
ZeroPoleToTransferFun(num,den,z,p,k);
//Define the frequency grid (logarithmic)
FreqFr->Length = 1000;
LogRamp(FreqFr,-1,1); //between 0.1 (=10^(-1)) and 10 (=10^1) rad/sec
FrequencyResponseS(num,den,FreqFr,Response); //Laplace
DrawIt(Response); //Y axis linear, X axis logarithmic
}
using Dew.Math;
using Dew.Math.Editors;
using Dew.Math.Units;
using Dew.Signal;
using Dew.Signal.Units;
using Dew.Math.Tee;
using Dew.Signal.Tee;
private void button1_Click(object sender, EventArgs e)
{
Vector z = new Vector(0);
Vector p = new Vector(0);
Vector num = new Vector(0);
Vector den = new Vector(0);
Vector Response = new Vector(0);
Vector FreqFr = new Vector(0);
double k,Wc,BW;
double FS = 2;
int Order = 5; //design a fifth order filter.
IIRFilters.EllipticAnalog(Order,0.2,40,z,p,out k); //design analog protype
Wc = Math.Sqrt(0.2*0.6);
BW = 0.6 - 0.2;
LinearSystems.LowpassToBandpass(z,p,ref k,Wc,BW); //frequency transformation in s-domain
LinearSystems.ZeroPoleToTransferFun(num,den,z,p,k);
//Define the frequency grid (logarithmic)
FreqFr.Length = 1000;
SignalUtils.LogRamp(FreqFr,-1,1); //between 0.1 (=10^(-1)) and 10 (=10^1) rad/sec
SignalUtils.FrequencyResponseS(num,den,FreqFr,Response,0); //Laplace
TeeChart.DrawIt(Response,"Frequency response",false); //Y axis linear, X axis logarithmic
}
Declaration
Procedure LowpassToBandpass(z, p: TVec; var k: TSample; CenterFreq, Bw: TSample);
Description
Transform a lowpass filter prototype in zero-pole form to bandpass filter, where the passband of width BW is centered around the frequency CenterFreq. Assumed sampling frequency is 2. The transformation is defined as ([1], p. 258):
s^2 + Wu*Wl
s --> -------------
s(Wu - Wl)
Wl - lower cutoff frequency
Wu - upper cutoff frequency
The routine also adds zeros at 0. It adds one zero, if the lowpass filter order is odd and already has zeros. If the filter does not have zeros, it adds sufficient zeros at 0 to match the order of the lowpass filter.